# Math Lesson Plan

Jessica Schrage

Created on September 18, 2024

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## Transcript

and Problem Solving

Patterns

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MATH 143

Finding a Pattern

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- Patterns play a major role in the solution of problems in all areas of life.
- Finding a pattern is such a useful problem-solving strategy in mathematics that some have called it the art of mathematics.
- To find patterns, we need to compare and contrast. We must compare to find features that remain constant and contrast to find those that are changing.

Example

Consider the sequence of figures. Find a pattern and determine the next figure.

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Patterns in Nature

A variety of patterns occur in plants and tress. Many of these patterns are related to a famous sequence of numbers called the Fibonacci Numbers. After the first two numbers of this sequence, which are 1 and 1, each successive number can be obtained by adding the two previous numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

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Pascal's Triangle

- The triangular pattern of numbers shown in the figure to the right is pascal's triangle. It has been of interest to mathematicians for hundreds of years, appearing in China, as early as 1303.
- This triangle is named after the French mathematician Blaise Pascal (1623-1662).
- In the fourth row, each of the numbers 4, 6, and 4 can be obtained by adding the two adjacent numbers from the row above it.
- Except for row 0, the second number in each row is the number of the row.

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Pascal's Triangle

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Arithmetic Sequence

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- The sequence 1, 2, 3, 4, 5, ....and 2, 4, 6, 8, 10,...are among the first that children learn.
- In such sequences, each new number is obtained from the previous number in the sequence by adding a selected number throughout.
- This selected number is called the common difference, and the sequence is called an arithmetic sequence.

Example

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Arithmetic Sequence: 7, 11, 15, 19, 23,...

Example

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Arithmetic Sequence: 7, 11, 15, 19, 23,...

Geometirc Sequence

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- In a geometric sequence, each new number is obtained by multiplying the previous number by a selected number.
- This selected number is called the common ratio, and the resulting sequence is called a geometric sequence.

Example

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Geometric Sequence: 1, 5, 25, 125, 625,...

Example

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Geometric Sequence: 1, 5, 25, 125, 625,...

Triangular Numbers

- The sequence of numbers illustrated below is neither arithmetic nor geometric.
- These numbers are called triangular numbers because the arrangement of dots that is associated with each number.
- Since each triangular number is the sum of whole numbers beginning with 1, the formula for the sum of consecutive whole numbers can be used to obtain triangular numbers (1, 1 + 2, 1 + 2 + 3, etc.).

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Finite Differences

There are other types of numbers that recieve their names from the number of dots in geometric figures. Such numbers are called figurate numbers, and they represent one kind of link between geometry and arithmetic.

Figurate Numbers

Often sequences of numbers don't appear to have a pattern. However, sometimes number patterns can be found by looking at the difference between consecutive terms. This approach is called the method of finite differences.

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Example

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- Consider the sequence:

- Find a pattern and determine the next term.
- Using the method of finite differences, we can obtain a second sequence of numbers by computing the differences between numbers from the original sequence.
- Then a third sequence is obtained by computing the differences from the second sequence. The process stops when all the numbers in the sequence of differences are equal.
- In this example, when the sequence becomes all 2's, we stop and work our way back from the bottom row to the original sequence.

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Inductive Reasoning

- The process of forming conclusions on the basis of patterns, observations, examples, or experiments is called inductive reasoning.
- The National Council of Teachers of Mathematics describes the type of reasoning as "Identifying patterns is a powerful problem-solving strategy. It is also the essence of inductive reasoning. As students explore problem situations apporpaite to their grade level, they can often consider or generate a set of specific instances, orgniaze them, and look for a pattern. These, in turn, can lead to conjectures about the problem.